weak topology in nLab

> For the *strong topology* in functional analysis, see at [[strong operator topology]].


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### Context
#### Topology
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[[!include topology - contents]]
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# Induced topologies
* table of contents
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## Definitions

See also at *[[Top]]* the section *[Universal constructions](Top#UniversalConstructions)*.


### Weak/coarse/initial topology
 {#WeakTopologyDefinition}

Suppose 

1. $S$ is a [[set]], 

1. $\{ (X_i, T_i) \}_{i \in I}$ is a [[family]] of [[topological spaces]] 

1. $\big\{ f_i \colon S \to X_i \big\}_{i \in I}$ an [[indexed set]] of [[functions]] from $S$ to the family $\{ X_i \}_{i \in I}$. 

Let $\Gamma$ denote the set of all topologies $\tau$ on $S$ such that $f_i$ is a [[continuous map]] for every $i \in I$. Then the [[intersection]] $\bigcap_{\tau \in \Gamma} \tau$ is again a topology and also belongs to $\Gamma$. Clearly, it is the coarsest/weakest topology $\tau_0$ on $X$ such that each function $f_i\colon S \to X_i$ is a [[continuous map]].

We call $\tau_0$ the **weak/coarse/initial topology induced** on $S$ by the family of mappings $\{ f_i \}_{i \in I}$.  Note that all terms 'weak topology', 'initial topology', and 'induced topology' are used.  The [[subspace topology]] is a special case, where $I$ is a [[singleton]] and the unique function $f_i$ is an [[injection]].



### Strong/fine/final topology
 {#StrongTopologyDefinition}

[[formal duality|Dually]], suppose 

1. $S$ is a [[set]], 

1. $\{ (X_i, T_i) \}_{i \in I}$ a [[family]] of [[topological spaces]] 

1. $\big\{ f_i \colon X_i \to S \big\}_{i \in I}$ a family of [[functions]] to $S$ from the family $\{ X_i \}_{i \in I}$. 

Let $\Gamma$ denote the set of all topologies $\tau$ on $S$ such that $f_i$ is a [[continuous map]] for every $i \in I$. Then the [[union]] $\bigcup_{\tau \in \Gamma} \tau$ is again a topology and also belongs to $\Gamma$. Clearly, it is the finest/strongest topology $\tau_0$ on $S$ such that each function $f_i\colon X_i \to S$ is a [[continuous map]].

We call $\tau_0$ the **strong/fine/final topology induced** on $S$ by the family of mappings $\{ f_i \}_{i \in I}$.  Note that all terms 'strong topology', 'final topology', and 'induced topology' are used.  The [[quotient topology]] is a special case, where $I$ is a [[singleton]] and the unique function $f_i$ is a [[surjection]].


## Generalisations

We can perform the first construction in any [[topological concrete category]], where it is a special case of an [[initial structure]] for a [[sink|source or cosink]].

We can also perform the second construction in any [[topological concrete category]], where it is a special case of a [[final structure]] for a [[sink]].

## Examples

### In functional analysis
 {#InFunctionalAnalysis}

In [[functional analysis]], the term 'weak topology' is used by default for the [[weak topologies of topological vector spaces]] (see there) with respect to their [[linear functionals]]: 

For if $V$ is a [[topological vector space]] over the [[ground field]] $K$, then we may consider the [[continuous linear functional]]s on $V$, that is the [[continuous map|continuous]] [[linear maps]] from $V$ to $K$.  Taking $V$ to be the set $X$ in the general definition above, taking each $T_i$ to be $K$, and taking the continuous linear functionals on $V$ to comprise the family of functions, then we get the __weak topology__ on $V$.

The _weak-star topology_ on the dual space $V^*$ of continuous linear functionals on $V$ is precisely the weak topology induced by the dual ([[evaluation]]) functionals on $V^*$

$$
  \left\{
    \array{
      V^* 
        &\overset{\operatorname{ev}_v}{\longrightarrow} & 
      K
      \\
      f 
        &\mapsto& 
      f(v)
    }
  \right\}_{v \in V}
  \mathrlap{\,.}
$$

For the strong topology in functional analysis, see at *[[strong operator topology]]*.


## Related entries

* [Category of topological spaces -- Universal constructions](Top#UniversalConstructions)

## References

The original version of this article was posted by [[Vishal Lama]] at [[nlabmeta:induced topology]].

See also

* Wikipedia, _[Initial topology](https://en.wikipedia.org/wiki/Initial_topology)_, _[Final topology](https://en.wikipedia.org/wiki/Final_topology)_, 


[[!redirects weak topology]]
[[!redirects weak topologies]]

[[!redirects coarse topology]]
[[!redirects coarse topologies]]

[[!redirects coarsest topology]]
[[!redirects coarsest topologies]]


[[!redirects initial topology]]
[[!redirects initial topologies]]

[[!redirects strong topology]]
[[!redirects strong topologies]]
[[!redirects fine topology]]
[[!redirects fine topologies]]
[[!redirects final topology]]
[[!redirects final topologies]]
[[!redirects weak-star topology]]
[[!redirects induced topology]]
[[!redirects induced topologies]]